In a prior blog post, I presented an uncommon method for solving the well-known Burning Tent problem. My solution, modeled on the approach in the Connected Geometry curriculum, used a dynamic ellipse to pinpoint the optimal solution. Now, I’d like to offer a related problem from Connected Geometry where the pedagogical benefit of using an ellipse becomes even clearer. Here is the situation:
You (at point Y) and your friend (at point F) are floating on inflatable lounge chairs in a circular swimming pool. You want to paddle to the edge of the pool to leave your sunglasses there, and then swim to your friend. Where should you paddle to minimize the total distance?
Page 1 of the Web Sketchpad model below shows the pool as well as points Y and F. Point P is a possible location where you might swim. You can drag point P around the circumference of the pool.
Notice that the model also includes an ellipse. The ellipse has foci at points Y and F and passes through point P. All the points on the ellipse are locations that are equally good (or bad) as swimming to point P.
Drag point P around the pool and watch as the ellipse grows and shrinks. Can you identify some interesting locations for P? In particular, notice when the ellipse is tangent to the circle. What do you think makes those locations noteworthy?
Now go to page 2 of the websketch (using the arrow in the lower-right corner). You’ll see the same swimming pool, but now there’s a graph showing the location of point P on the pool’s circumference plotted against the total distance to swim. As you drag point P, notice the locations where the ellipse is tangent to the circle and how those correspond to significant points on the graph.
This exploration introduces notions from calculus such as absolute minimum, absolute maximum, local minimum, and local maximum all without any algebra or reliance on derivatives. For example, when point P sits at the local minimum of the graph, the ellipse is tangent to the circle, with those points to the right or left of P on the pool’s circumference sitting outside the ellipse (see the picture below). Thus point P is a better place to swim than any of those nearby points. However, point P is not the absolute best place to swim because there are locations on the other side of the pool that are inside the ellipse. By examining the ellipse in relation to the pool’s circumference, students can explain why a location of point P is a absolute minimum, absolute maximum, local minimum, or local maximum without even seeing the graph.
You can change the graph by dragging points Y and F to new locations. Can points Y and F be positioned so that there are: (i) two locations of point P that produce the absolute minimum? (ii) no locations of point P that produce the absolute minimum? (iii) no locations of point P that produce a local minimum?
On page 3 of the websketch, the pool is now rectangular, leading to an even more interesting graph.